A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations

0.38 MB·English

by  David Kinderlehrer

A WASSERSTEIN GRADIENT FLOW APPROACH TO
POISSON-NERNST-PLANCK EQUATIONS
DAVIDKINDERLEHRER,LÉONARDMONSAINGEON,AND XIANGXU
5
1
0 Abstract. ThePoisson-Nernst-Planck system of equationsusedtomodelionic
2 transportisinterpretedasagradientflowfortheWassersteindistanceandafree
energy in the space of probability measures with finitesecond moment. A varia-
p
tionalschemeisthensetupandisthestartingpointoftheconstructionofglobal
e
S weak solutions in a unified framework for the cases of both linear and nonlinear
diffusion. The proof of the main results relies on the derivation of additional es-
6
timatesbasedontheflowinterchangetechniquedevelopedbyMatthes,McCann,

and Savaréin [25].
]
P
A
Contents
.
h
t 1. Introduction 1
a
2. Formal Wasserstein gradient flow 5
m
3. Study of the energy functionals 9
[
4. Minimizing scheme and discrete estimates 16

3
5. Convergence to a weak solution 20
v
6. Appendix 28
7
3 References 29
4
4
0
.
1 1. Introduction
0
5 The Poisson-Nernst-Planck (PNP) system of equations [18, 24] is the principal
1
description of ionic transport of several interacting species. It has been applied in
:
v a number of contexts ranging from electrical storage devices to molecular biology,
i
X at times coupled to Navier-Stokes or other systems. The basic system is to find
r u(t,x) ≥ 0 and v(t,x) ≥ 0 satisfying
a
∂ u= ∆um+div(u∇(U +ψ)),
t
∂ v = ∆vm+div(v∇(V −ψ)), t ≥ 0, x ∈ Rd, d≥ 3, (1.1)
t
−∆ψ = u−v,
for some suitable initial conditions u| =u0 and v| = v0. The unknowns u and
t=0 t=0
v represent the density of some positively and negatively charged particles. Here
m ≥ 1 is a chosen fixed nonlinear diffusion exponent. Note that (1.1) formally
preserves the L1 mass
u(t,x)dx = u0(x)dx and v(t,x)dx = v0(x)dx
ZRd ZRd ZRd ZRd
for all t ≥ 0, which physically represents the conservation of total charge of the
individualspecies. Forinitialmasses u0(x)dx = v0(x)dx,asimplerescalingof
Rd Rd
timeandspaceallowsnormalizationoRfmassestouniRty, andwithoutlossofgenerality
1
2
we consider u(t,x),v(t,x) to be probability densities. The external potentials U(x)
and V(x) are prescribed and sufficiently smooth. ψ(x) from the Gauss Law is the
self-consistent electrostatic potential created by the two charge carriers according to
the last equation in (1.1). The first two equations in (1.1) are called Nernst-Planck
equations and describe electro-diffusion and electrophoresis according to the Fick
and Kohlrausch laws, respectively, while the last equation in (1.1) corresponds to
the electrostatic Poisson law. The boundary condition for this coupling equation
will always be understood in the sense of the Newtonian potential: we shall always
implicitly write
−∆ψ = u−v ⇔ ψ = G∗(u−v) = (−∆)−1(u−v), (1.2)
where
1
G(x) := |x|2−d
d(d−2)ω
d
is the fundamental solution of −∆ in Rd and ω the volume of the unit ball in Rd
d
(for d ≥ 3).
In some PNP models, an extra background doping profile C(x) is considered
resulting in the modified potential −∆ψ′ = u−v +C. With suitable assumptions
on C(x) this can be easily eliminated replacing the external potentials U by U+ψ
C
and V by V −ψ , where ψ = (−∆)−1C.
C C
There is a vast literature on well-posedness and long time behavior of the system
(1.1). We refer to [6, 13, 14, 17] and references therein for bounded domains, and
[3, 4, 7, 12, 21] for the whole space problem. Different from these papers, our
contribution is to show that the system of equations (1.1), governing drift, diffusion
and reaction of charged species, possesses a gradient flow structure as viewed on the
metric space of probability measures on Rd endowed with the quadratic Wasserstein
distance, inessencetheweak-∗topology(seealso[28,31,36]forvariousdiscussions).
Therefore, variational methods may be introduced to prove global existence of weak
solutions (see definition below). Motivation for this in terms of energy dissipation
is offered below. Also, we provide a unified framework both for linear m = 1 and
nonlinear m > 1 diffusions. To the best of our knowledge well-posedness in the
whole space for m > 1 usually requires either high integrability of the initial data
or initial gradient regularity. We would like to stress that we need here no such
hypotheses and that our result merely requires some low initial integrability, defined
in terms of the diffusion exponent m ≥ 1 and dimension d≥ 3 only (which we think
is sharp, see discussion after the proof of Proposition 4.4 page 19). Also, we do not
need compatibility conditions between m ≥ 1 and d ≥ 3. This is in contrast with
the Patlak-Keller-Segel chemotaxis models [8, 9], for which critical mass phenomena
may occur depending on whether m is larger or smaller than the scale-invariant
exponent m (d) = 2− 2, see e.g. [10] for the critical parabolic-parabolic case. This
∗ d
ismainlyduetothefactthattheself-induceddriftsarerepulsivehere,whiletheyare
self-attractive in the Keller-Segel models, thus leading to aggregation and blow-up
in finite time.
It was shown in [16, 30] that certain scalar diffusion equations can be interpreted
as gradient flows in metric spaces and the literature concerning this issue is steadily
growing (see [2] and references therein). It is thus a question of great interest to
apply such ideas to study systems of equations. In contrast, there are only a few
examples for systems. For related studies, we refer to [8, 9, 10, 15, 20, 22, 26, 37],
where the energy functional is involved with the Wasserstein metric and existence
3
theoremsusingaminimizingmovementschemeforcorrespondingevolutionproblems
are presented.
In [16], the linear Fokker-Planck equation
∂ ρ = σ∆ρ+div(ρ∇ϕ)
t
is regarded as the gradient flow of a free energy consisting of the Boltzmann entropy
with a potential ϕ,
F(ρ) = (ϕρ+σρlogρ)dx,
ZRd
with respect to the quadratic Wasserstein metric. There may be many Lyapunov
functions associated to a differential equation. The result of [16] means that dis-
sipation for the free energy F determines the Fokker-Planck Equation. The same
idea was later employed in [30] and, in addition, to derive long-time asymptotics for
the Porous Medium Equation (PME). Since the system (1.1) can be viewed as two
Fokker-Planck equations (when m = 1) or Porous Medium equations (when m > 1)
in u,v coupled by means of a Poisson kernel, we are motivated to extend these ideas
to study our coupled system. Inspired by [30] and [35], we shall in fact discover that
the PNP system can be seen as a gradient flow driven by the free energy
|∇ψ|2
E(u,v) := ulogu+vlogv+uU +vV + dx if m = 1,
ZRd(cid:16) 2 (cid:17)
(1.3)
um vm |∇ψ|2
E(u,v) := + +uU +vV + dx if m > 1.
ZRd(cid:16)m−1 m−1 2 (cid:17)
We further motivate this approach informally by discussing the relationship be-
tween the dissipation relation and the weak-∗ topology in terms of the Wasserstein-
Rubinstein-Kantorovich distance, orsimply the Wassersteindistance. Wefollow [36]
and consider for illustration the case m = 1. Set
1
ϕ(u,v) = ulogu+vlogv+uU +vV + |∇ψ|2, so that
2
E(u,v) = ϕ(u,v)dx.
ZRd
Given a process or an evolution (u(t),v(t)), during an interval (T,T +h) the change
in energy is
T+h d
E(u,v) − ϕ(u,v)dxdt = E(u,v) (1.4)
(cid:12)T+h ZT ZRd dt (cid:12)T
(cid:12) (cid:12)
This, (1.4), is the dissi(cid:12)pation equality or inequality and the den(cid:12)sity of the middle
term
d
D = − ϕ(u,v)dx (1.5)
ZRd dt
is the dissipation density along the trajectory. Writing
d du dv
ϕ(u,v) =ϕ +ϕ , (1.6)
u v
dt dt dt
we must ascribe a meaning to
du dv
,
dt dt
to render the system dissipative, that is, so that (1.5) is positive. To begin we
calculate the terms in (1.6). Keeping in mind (1.2), one checks that
δ 1 δ 1
|∇ψ|2 = ψ and |∇ψ|2 = −ψ,
δu 2 δv 2
(cid:16) (cid:17) (cid:16) (cid:17)
4
which leads to
ϕ (u,v) = logu+U +ψ+1 and ϕ = logv+V −ψ+1.
u v
Let us now employ the Poisson-Nernst-Planck equations (1.1). Substituting into
(1.5) and integrating by parts gives
∇u 2 ∇v 2
D = +∇(U +ψ) u+ +∇(V −ψ) v dx
ZRd(cid:26)(cid:12) u (cid:12) (cid:12) v (cid:12) (cid:27)
(cid:12) (cid:12) (cid:12) (cid:12)
Introduce (cid:12) (cid:12) (cid:12) (cid:12)
∇u ∇v
w =− +∇(U +ψ) and ω = − +∇(V −ψ) so that
u v (1.7)
(cid:16) (cid:17) (cid:16) (cid:17)
u +div(wu) = 0 and v +div(ωv) =0.
t t
We have that
T+h T+h T+h
Ddt = |w|2udxdt+ |ω|2vdxdt
ZT ZT ZRd ZT ZRd
where the pairs (u,w),(v,ω) satisfy the continuity equations (1.7). This represents
a trial in d , the quadratic Wasserstein metric, using the Benamou-Brenier formu-
W
lation [5], where
1 T+h
d (u ,u )2 = inf |w|2udxdt and
h W (cid:12)T+h (cid:12)T ZT ZRd
1 (cid:12) (cid:12) T+h
d (v ,v )2 =inf |ω|2vdxdt
h W (cid:12)T+h (cid:12)T ZT ZRd
taken o(cid:12)ver all(cid:12)pairs (u,w) and (v,ω) satisfying the continuity equations
u +div(wu) = 0 and v +div(ωv) =0, and
t t
the initial and terminal conditions.
The calculation shows that the dissipation relation (1.7) for E and the PNP system
are closely related to the Wasserstein distance. We could write, in fact,
1 1
d (u ,u )2+ d (v ,v )2+E(u,v) ≤ E(u,v)
h W T+h T h W T+h T T+h T
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
suggesting an im(cid:12)plicit s(cid:12)cheme which l(cid:12)eads to(cid:12)a gradient flo(cid:12)w. This is nea(cid:12)rly correct.
As is well known, a factor of 1/2 must be inserted; see below (1.8).
We now turn to the precise formulation. Denoting P(Rd) the set of Borel proba-
bility measures on Rd with finite second moments and d the quadratic Wasserstein
W
distance as before, the underlying space will be here (u,v) ∈ P(Rd) ×P(Rd) and
will inherit a natural differential structure from that of (P(Rd),d ) - see section 2
W
for details. The total free energy (1.3) is a combination of the well-known internal
(diffusive entropy) and potential energies for each species, and, although unclear
at this stage, the coupling Dirichlet energy 1 |∇ψ|2dx falls into the category of
2 Rd
so-called interaction energies. See [35] for an iRntroduction.
Following [16], we shall construct weak solutions z = (u,v) as follows. Given
suitable initial data z0 = (u0,v0) and some small time step h ∈ (0,1) we first
(n)
construct a discrete sequence {zh }n∈N solution to the Jordan-Kinderlehrer-Otto or
JKO implicit scheme
z(0) = z0,
h
1 (1.8)
z(n+1) ∈ Argmin d2 ·,z(n) +E(·)
h (cid:26)2h h (cid:27)
K×K (cid:0) (cid:1)
Here E(z) = E(u,v) is the total free energy (1.3), d2 is the (squared) distance5
on the product space inherited from d , and K ⊂ P the set of admissible mini-
W
mizers defined later on. As is classical by now, one obtains interpolating solutions
{z (t)} = {u (t),z (t)} defined for all t ≥ 0, piecewise constant in time, and sat-
h h h h h
isfying a coupled system of two Euler-Lagrange equations. We shall then prove that
as h → 0 one recovers a weak solution (u(t),v(t)) = z(t) = lim z (t) of (1.1). There
h
h→0
are several challenges in this program.
In handling the |∇ψ|2dx coupling term, some intrinsic difficulties arise due
Rd
both to the specificRPoisson kernel and to the nonscalar setting. First, as neither the
external potentials nor G are convex the free energy E is not displacement convex in
the sense of McCann [27] and we cannot simply apply the standard procedures as
in [2]. Secondly, due to the singular nature of the kernel G(x) = C/|x|d−2 both the
existence of minimizers in (1.8) and derivation of the corresponding Euler-Lagrange
equations become delicate, see in particular the discussions in Proposition 3.1 and
Proposition4.4fordetails. Inordertotacklethisissueweusedthe"flowinterchange"
technique that originates in [25] and was later used in [10, 22] to obtain some inte-
grability improvement and gradient regularity of the minimizers, see Proposition 3.4
and Proposition 3.5 below. The highlight of the argument is the propagation of ini-
tialLp(Rd)regularity establishedinProposition 3.4. Inadditiontobeingtechnically
essential here, this propagation of initial regularity allowed us to obtain a natural
L∞(Rd) estimate (see Theorems 2-3), which to the best of our knowledge was un-
known for the PNP system in the whole space. We also believe that the very same
argument could be employed for similar problems in order to show propagation of
initial regularity, which is usually a delicate point in the mass transport framework.
The rest of the paper is organized as follows. In Section 2 we recall well-known
facts in optimal transport theory and briefly describe the differential structure of
the product space. We then formally derive the Wasserstein gradient flow structure
of the system (1.1) and state the main existence results. In Section 3 we study the
relevantenergyfunctionals, andestablishimprovedregularity oftheirminimizers. In
Section 4wefix atimesteph> 0smallenough andconsider theminimizing scheme.
We obtain approximate discrete solutions {u ,v } and derive the corresponding
h h h
Euler-Lagrange equations. In Section 5 we take the limit h → 0 and show that the
convergence (u,v) = lim(u ,v ) is strong enough to retrieve a weak solution. This
h h
h→0
last section also contains the proof of the main theorems.
Notation Convention. Unlessotherwisespecified, h·,·iand·denoteinnerproduct
of elements in Rd, P denotes P(Rd), and Pac denotes Pac(Rd). If clear from the
context we shall often omit the subscripts m = 1 or m > 1. If 1 ≤ p ≤ ∞, we denote
by p′ = p the conjugate Lebesgue exponent.
p−1
2. Formal Wasserstein gradient flow
From now on we assume that the external potentials are quadratic at infinity, i-e
C |x|2 ≤U(x),V(x) ≤ C (1+|x|2)
1 2
|∇U(x)|,|∇V(x)| ≤ C (1+|x|) (2.1)
3
k∆UkL∞(Rd),k∆VkL∞(Rd) ≤ C4,
for somegeneric positive constants C . Notethat U,V need not be uniformly convex
i
asisoftenassumed,sothatweallowheremultiplewells. Althoughtheseassumptions
6
on the potentials could be weakened we assume here strict confinement C > 0 for
1
the ease of exposition, and we do not seek optimal generality.
We also introduce the admissible set
K := Pac(Rd)∩L1logL1(Rd) if m = 1,
K := 1 (2.2)
(cid:26) Km := Pac(Rd)∩Lm(Rd) if m > 1,
and for reasons that shall become clear later on we shall always consider initial data
u0,v0 ∈ K∩Lr0(Rd) for some r > max{m,2d/(d+1)}. (2.3)
0
Essentiallyu0,v0 ∈ Lm∩L2d/(d+2)(Rd)ensuresthattheinitialenergy(u0,v0)isfinite,
while L2d/(d+1)(Rd) regularity will ensure that the self-induced drifts u∇Ψ,v∇Ψ ∈
L1(Rd) for all times. We believe that r = max{m,2d/(d+1)} should be admissible,
but for technical compactness issues we have to assume here slightly better Lr0(Rd)
integrability for some r > r arbitrarily close. See the proof of Theorem 4 later on
0
for details.
For ρ,u,v ≥ 0 let us define the usual Boltzmann entropy
H(ρ):= ρlogρdx,
ZRd
the diffusion energy
E (u,v) := ulogu+vlogv dx if m =1 and
diff
ZRd(cid:0) (cid:1) (2.4)
1
E (u,v) := um+vm dx if m > 1
diff
m−1ZRd(cid:0) (cid:1)
and the external potential energy
E (u,v) := (uU +vV)dx. (2.5)
ext
ZRd
Note that, with our assumptions, E ,E are finite for all (u,v) ∈ K × K. For
diff ext
ψ = (−∆)−1(u−v) = G∗(u−v) we define now the coupling energy
1
E (u,v) := |∇ψ|2dx, (2.6)
cpl
2 ZRd
whichistheenergyoftheself-inducedelectricpotential. Notethat,atleastformally,
|∇ψ|2dx= (−∆ψ)ψdx = (u−v)G∗(u−v)dx
ZRd ZRd ZRd
= [u−v](x)G(x−y)[u−v](y)dxdy (2.7)
ZZRd×Rd
falls into the category of interaction energies
ρ(x)K(x,y)ρ(y)dxdy
ZZRd×Rd
treated in [35]. To sum up, the total free energy E = E +E +E is given by
diff ext cpl
(1.3)
For given probability measures µ,ν ∈ P(Rd) we denote the squared (quadratic)
Wasserstein distance by
d2 (µ,ν) = inf |x−y|2dγ(x,y),
W
γ∈Γ(µ,ν)ZZRd×Rd
where Γ(µ,ν) ⊂ P(Rd×Rd) is the set of admissible joint distributions with x and y
x y
marginals µ,ν respectively. We recall from [35] that (P,d ) is a metric space and
W
that d metrizes the weak convergence of measures. When µ ∈ Pac is moreover7
W 2
absolutely continuous with respect to the Lebesgue measure dµ(x)≪ dx, the square
Wasserstein distance can also be computed by the Benamou-Brenier theorem [5] as
already noted.
Theorem 1 (Existence of optimal maps, [35]). Let µ ∈ Pac(Rd) and ν ∈ Pac(Rd).
2 2
There exists a unique optimal transport map T = ∇ϕ ∈ L2(Rd;dµ) for some convex
function ϕ such that
ν = T µ : ∀f ∈ C (Rd), f(y)dν(y)= f ◦T(x)dµ(x)
# c
ZRd ZRd
and
d2 (µ,ν) = |x−T(x)|2dµ(x).
W
ZRd
Our interest here is a system so we endow P(Rd)×P(Rd) with the natural product
distance
d2(z,z′)= d2 (u,u′)+d2 (v,v′)
W W
for all z = (u,v) and z′ = (u′,v′) in P ×P. It is well known [30, 35] that (P,d )
W
enjoys a natural differential structure defined by means of continuity equations, so
that our product space also has the same differential structure. This permits us to
differentiate real-valued functions F on the product space, and defines the corre-
sponding Wasserstein gradient by the chain rule dF(z ) = grad F(z ).dzt. We
dt t W t dt
show now that (1.1) is really the gradient flow
dz u(t)
= −grad E(z), z(t) = .
dt W v(t)
(cid:16) (cid:17)
Intermsofordinary calculusofvariations, werecallthatthisisachievedbyvariation
of domain or, in fluid dynamical terms, by Lagrangian variations, [16]. To this end,
let us split for convenience the coupling Poisson equation −∆ψ = u−v as
−∆ψ = u ⇔ ψ = G∗u,
ψ = ψ −ψ with u u
u v (cid:26) −∆ψv = v ⇔ ψv = G∗v.
Formally differentiating the diffusive energy (2.4) with respect to u (resp. v), a by
now classical computation [30, 35] leads to the ∆um (resp. ∆vm) term in (1.1).
Similarly, differentiating the external energy (2.5) with respect to u and v classically
gives rise to ∇·(u∇U) and ∇·(v∇V) in (1.1). In order to differentiate the coupling
termwefirstusetheformalintegration byparts(2.7)andthenexploitthesymmetry
G(x−y)= G(y−x) to expand
1
E (u,v) = [u−v](x)G(x−y)[u−v](y)dxdy
cpl
2ZZRd×Rd
1
= u(G∗u)+v(G∗v) dx− u(x)G(x−y)v(y)dxdy.
2ZRd(cid:2) (cid:3) ZZRd×Rd
Differentiating with respect to u, it is well known [35] that the first integral gives
the corresponding ∇·(u∇(G∗u))+0 = ∇·(u∇ψ ) term. Rewriting the remaining
u
cross term
− uGvdx = − u(G∗v)dx = − uψ dx,
v
ZZRd×Rd ZRd ZRd
and noting that ψ is independent of u, it is again well known that this term gives
v
rise to −∇·(u∇ψ ). Summing up we obtain ∇·(u∇(ψ −ψ )) = ∇·(u∇ψ) as in
v u v
8
the first equation of (1.1). Similarly differentiating with respect to v we obtain the
−∇·(v∇ψ) term appearing in the second component.
Though very general notions of solutions related to Energy Dissipation Equality
(EDE) or Evolution Variational Inequality (EVI) can be used for abstract gradient
flows in metric spaces, [1, 2], we use the more direct framework, introducing some
features later for the implementation of the flow-interchange method.
Definition 2.1. A pair u,v : (0,∞)×Rd → R+ is a global weak solution if u,v ∈
C([0,∞);P), u(t),v(t) → u0,v0 in (P,d ) as t ց 0, ∇um, ∇vm, u∇U, v∇V, u∇ψ
W
and v∇ψ ∈ L2(0,T;L1(Rd)) for all T > 0, and for any fixed ϕ ∈ C∞(Rd)
c
d
u(t,x)ϕdx = − h∇um(t,x),∇ϕidx− u(t,x)h∇U,∇ϕidx
dtZRd ZRd ZRd
− u(t,x)h∇ψ(t,x),∇ϕidx, (2.8)
ZRd
and
d
v(t,x)ϕdx = − h∇vm(t,x),∇ϕidx− v(t,x)h∇V,∇ϕidx
dtZRd ZRd ZRd
+ v(t,x)h∇ψ(t,x),∇ϕidx (2.9)
ZRd
hold in the sense of distributions D′(0,∞) with ψ(t,x) = G∗[u−v](t,x) a.e. (t,x) ∈
(0,∞)×Rd.
WeobservethatL2 ([0,∞);L1(Rd))could bereplaced by L1 ((0,∞)×Rd)inthe
loc loc
above definition, which is enough for all the integrals in (2.8)-(2.9)to make sense. In
any case the weak solutions constructed here would still enjoy strong regularity in
the end, and our choice of including the regularity in the definition of weak solutions
is purely practical.
In this setting our main result is
Theorem 2 (Existence of solutions for m > 1). Fix m > 1 and initial data u0,v0
as in (2.3). Then there exists a global weak solution (u,v) with
u,v ∈ L∞ 0,∞;Lm(Rd)∩L1(Rd,(1+|x|2)dx)
∇ψ ∈ L∞((cid:0)0,∞;L2(Rd)) (cid:1)
(2.10)
um2 ,vm2 ∈L2(0,T;H1(Rd))
u,v ∈ L∞(0,T;Lp(Rd)), ∀p∈ [1,2d/(d+1)],
for all T > 0, and
E(u(t),v(t)) ≤ E(u0,v0) for a.e. t ≥ 0. (2.11)
If we further assume that u0,v0 ∈ Lp(Rd) for some p ∈[1,∞], then ∀τ ≥ 0,
sup ku(t)k +kv(t)k ≤ Ceλτ ku0k +kv0k , (2.12)
Lp(Rd) Lp(Rd) Lp(Rd) Lp(Rd)
t∈[0,τ](cid:16) (cid:17) (cid:16) (cid:17)
with
λ = max k∆UkL∞(Rd),k∆VkL∞(Rd) . (2.13)
(cid:8) (cid:9)
In the case of linear diffusion we have similarly
9
Theorem 3 (Existence of solutions for m = 1). The conclusions of Theorem 2 hold
for m = 1 if we replace u,v ∈ L∞(0,∞;Lm(Rd)) in (2.10) by
u,v ∈ L∞(0,∞;L1logL1(Rd)).
We would like to stress again that estimate (2.12) holds for p = ∞, and was not
known in the whole space as far as we can tell. Since we interpret (1.1) as a gradient
flow one could expect energy monotonicity E(t) ↓. This would immediately follow
from (2.11) and uniqueness of solutions. Unfortunately due to the lack of regularity
and displacement convexity we were not able to prove uniqueness within the above
class of weak solutions, and therefore we only retrieve an energy upper bound.
It is worth mentioning that the gradient flow structure of the PNP system and
the above theorems are also valid in the bounded domain case, with some mild
assumptions on the boundary and minor modifications of the proofs. Suppose Ω ⊂
Rd is a smooth bounded and convex domain, and consider the physically relevant
boundary condition, that is, the no flux boundary condition
∂u ∂v ∂ψ
= = = 0 on ∂Ω, ψdx = 0,
∂ν ∂ν ∂ν Z
∂Ω
where ν is the unit outward normal on ∂Ω. We also assume the external potentials
satisfy
∂U ∂V
= = 0 on ∂Ω
∂ν ∂ν
By [19] we know that the electrostatic potential can be represented as
ψ(x) = N(x,y)(u−v)(y)dy, ∀x ∈ Ω.
Z
Ω
Here the (singular) kernel N(x,y) = N(y,x) serves as a counterpart of the Green’s
functionG(x−y)inRd fortheNewtonpotential. Thenwemayargueinasimilarway
thatthePNPsystemformallypossessesthegradientflowstructure,andtheexistence
theorems can be proved in a similar but somewhat technically easier manner.
3. Study of the energy functionals
In this section we study various properties of the two relevant energy functionals,
namely the total free energy E and the functional (3.3) used in the JKO minimizing
scheme. As already mentioned in the introduction, we use the flow interchange
technique to establish some improved regularity for the minimizers, which will turn
out to be crucial in the next sections.
ForfurtherusewerecallhereaparticularcaseofthecelebratedHardy-Littlewood-
Sobolev (HLS) inequality:
Lemma 3.1 (Hardy-Littlewood-Sobolev, [23, 33]). In dimension d ≥ 3 let w ∈
Lp(Rd) and Φ = G∗w. Then
(1) If 1< p < d/2 there is a C = C(p,d) such that
kΦk ≤Ckwk , (HLS-1)
dp Lp(Rd)
Ld−2p(Rd)
while if p = 1 there is a C = C(d) such that
kΦk ≤ Ckwk . (HLS-2)
d L1(Rd)
Lwd−2(Rd)
(2) If 1< p < d there is a C = C such that
p,d
k∇Φk ≤ Ckwk . (HLS-3)
dp Lp(Rd)
Ld−p(Rd)
1S0ince |G(x)| = C and |∇G(x)| = C this is a particular case of well known
|x|d−2 |x|d−1
fractional integration results for the Riesz potential I f = 1 ∗f with α = 2,1,
α |.|d−α
and we refer to [23, 33] for details. Here Lq(Rd) denotes the weak-Lq space and
w
coincides with the usual Lorentz space Lq,∞(Rd).
As an immediate consequence we have the following integration by parts formula:
Proposition 3.1. Let d ≥ 3 and w ∈ L2d/(d+2)(Rd). Then Φ = (−∆)−1w = G∗w
satisfies Φ ∈ L2d/(d−2)(Rd), ∇Φ∈ L2(Rd), and
|∇Φ|2dx = Φwdx = w(x)G(x−y)w(y)dxdy. (3.1)
ZRd ZRd ZZRd×Rd
We shall use this later on with w = u−v in order to control ψ = G∗(u−v).
Proof. Taking p = 2d ∈ (1,d/2) in (HLS-1)(HLS-3) we see that Φ ∈ L2d/(d−2)(Rd)
d+2
and ∇Φ ∈ L2(Rd). Since (2d/(d + 2))′ = 2d/(d −2) all the integrals in (3.1) are
absolutely convergent and the last equality holds by Fubini’s theorem. In order to
retrieve the first equality we use approximation: if w ∈ C∞(Rd) converges to w in
n c
L2d/(d+2)(Rd) then by the HLS lemma Φ → Φ in L2d/(d−2)(Rd) and ∇Φ → ∇Φ in
n n
L2(Rd). Since (3.1) holds for smooth w ∈ C∞ with −∆Φ = w we conclude by
n c n n
letting n → ∞. (cid:3)
Back to our energy functional, we begin with a fairly standard type of result
[8, 16]:
Proposition 3.2 (Energy lower bound). Let m ≥ 1 and K as in (2.2). The total
free energy E is a proper functional on K×K and
inf E(u,v) > −∞.
K×K
Moreover we have in every sub-levelset {E(u,v) ≤ R} that
(i) gradient control: k∇ψk ≤ C
L2(Rd)
(ii) no concentration: if m > 1 then
(um+vm)dx ≤ C,
ZRd
while if m = 1 then
(u|logu|+v|logv|)dx ≤C.
ZRd
(iii) mass confinement: |x|2(u+v)dx ≤ C,
Rd
for some C > 0 depending oRn R > 0, the confining potentials, and m.
Proof. Choosing u,v smooth and compactly supported it is clear that E(u,v) < ∞
so E is proper.
m> 1 : (i)-(ii) immediately hold because each term in (1.3) is nonnegative. (iii)
then follows by (uU +vV)dx ≤ E(u,v) together with (2.1).
Rd
m= 1 : if m (ρ)R= |x|2ρdx denotes the second moment let us first recall [16] the
2 Rd
Carleman estimateR
H(ρ) ≥ − ρ(logρ)−dx ≥ −C(1+m (ρ))α,ρ ∈ P, (3.2)
2
ZRd